Problem: $\overline{AB}$ = $52$ $\overline{BC} = {?}$ $A$ $C$ $B$ $52$ $?$ $ \sin( \angle BAC ) = \dfrac{5}{13}, \cos( \angle BAC ) = \dfrac{12}{13}, \tan( \angle BAC ) = \dfrac{5}{12}$
$\overline{AB}$ is the hypotenuse $\overline{BC}$ is opposite to $\angle BAC$ SOH CAH TOA We know the hypotenuse and need to solve for the opposite side so we can use the sine function (SOH) $ \sin( \angle BAC ) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{\overline{BC}}{\overline{AB}}= \frac{\overline{BC}}{52} $ Since we have already been given $\sin( \angle BAC )$ , we can set up a proportion to find $\overline{BC}$ $ \sin( \angle BAC ) = \dfrac{5}{13} = \frac{\overline{BC}}{52}$ Simplify. $\overline{BC} = 20$